A new centered Gaussian process

In this work, we introduce a new Gaussian process called the mixed bifractional Brownian motion, defined as a linear combination of a finite number of independent bifractional Brownian motions. This process extends the classical bifractional Brownian motion and provides a more flexible framework for modeling complex stochastic phenomena exhibiting memory and self-similarity. The construction allows the incorporation of different bifractional parameters, leading to a richer covariance structure and a wider range of potential applications.

We investigate several fundamental probabilistic and path properties of the mixed bifractional Brownian motion. In particular, we establish its Hölder continuity and self-similarity properties and study the regularity of its sample paths. We also analyze the differentiability of the trajectories and derive the behavior of its quadratic variation. Furthermore, we examine the dependence structure of the process by investigating its long-range dependence property and the stationarity of its increments. The Markov property is also discussed, and conditions under which it may hold are examined.

In addition, we study the noise generated by the increments of the mixed bifractional Brownian motion and analyze its asymptotic behavior. These results provide a deeper understanding of the stochastic dynamics associated with this new process. The proposed model combines important characteristics such as self-similarity, long-range dependence, and non-stationary increments, making it a promising candidate for applications in fields where such features naturally arise, including finance, telecommunications, hydrology, signal processing, and other areas involving complex random phenomena.